Nonparametric Adjoint-Based Inference for Stochastic Differential Equations

We develop a nonparametric method to infer the drift and diffusion functions of a stochastic differential equation. With this method, we can build predictive models starting with repeated time series and/or high-dimensional longitudinal data. Typical use of the method includes forecasting the future density or distribution of the variable being measured. The key innovation in our method stems from efficient algorithms to evaluate a likelihood function and its gradient. These algorithms do not rely on sampling, instead, they use repeated quadrature and the adjoint method to enable the inference to scale well as the dimensionality of the parameter vector grows. In simulated data tests, when the number of sample paths is large, the method does an excellent job of inferring drift functions close to the ground truth. We show that even when the method does not infer the drift function correctly, it still yields models with good predictive power. Finally, we apply the method to real data on hourly measurements of ground level ozone, showing that it is capable of reasonable results.